Mereology (parthood as colimit)
Parthood, in the τ-framework, is the canonical injection into a colimit. A whole W is composed of parts {Pᵢ} iff W = colim(D) for a diagram D in the admissible category whose objects are the parts. Composition occurs only when the colimit exists in the admissible category — not every plurality forms a whole, and the question 'is X a part of Y' has a structural answer.
τ-Definition
Parthood, in the τ-framework, is the canonical injection into a colimit. A whole W is composed of parts {Pᵢ} iff W = colim(D) for a diagram D in the admissible category whose objects are the parts. Composition occurs only when the colimit exists in the admissible category — not every plurality forms a whole, and the question 'is X a part of Y' has a structural answer.
Categorical invariant. P ⊑_τ W ⟺ P ∈ Obj(D) ∧ W = colim(D) ∧ canonical injection P → W exists in AdmCat.
Primary registry anchor:
VII.D35
τ-Derivation Chain
-
VII.D25— Internal Set Ontology — parts and wholes are NF-addressable. -
VII.D37— Six Ontic Requirements (OR1–OR6) — only admissible diagrams are eligible for colimit composition. -
VII.D35— Mereological Composition as Colimit — whole = colim(parts) when the colimit exists in AdmCat. -
VII.D34— Identity as Address Persistence — composite identity tracks the colimit's NF-address through morphisms.
Lean modules referenced:
TauLib.BookVII.Meta.Registers
Phenomenological Correlate
Mereology is instantiated whenever we ask 'is X part of Y' or 'what is Y made of'. A molecule is composed of its atoms iff the atom-diagram has an admissible colimit (the bonded structure); a sentence is composed of its words iff the word-diagram has an admissible colimit (the parsed expression); a committee is composed of its members iff the membership-diagram has an admissible colimit (the legitimately constituted body). Composition is not arbitrary — the admissibility constraint is what distinguishes a heap of letters from a word.
Examples:
- Empirical: 'water is H₂O' — the molecule is the colimit of the diagram {2 H atoms, 1 O atom, bonding morphisms} in the admissible chemistry category.
- Linguistic: a sentence is the colimit of its parsed-word diagram; an unparseable string fails to compose because no admissible colimit exists.
- Normative: a constituted committee is the colimit of its membership diagram in Reg_P; a 'committee' formed by illicit appointment fails admissibility and is not a whole.
- Mathematical: a topological space is the colimit of its open-cover diagram; sheaf-gluing is exactly colimit-existence in the relevant category.
Register codomain: Cross-register (mereological composition is defined wherever a register has a category with admissible colimits).
Manuscript reference: manuscript-sources/book-07/part02/ch27.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookVII.Meta.Registers
Lean kind: structure
Lean symbol: Tau.BookVII.Meta.Registers.MereologicalCompositionAsColimit